Heat oven to 350°F. Line 12 muffin cups with paper baking liners.
In food processor, pulse chopped persimmons until mostly smooth. Measure out 3/4 cup pulp; save or discard leftovers.
In large bowl, whisk flour, baking soda, 1/2 teaspoon cinnamon, nutmeg, salt, allspice and cloves. In separate large bowl or bowl of stand mixer, cream butter and sugar on medium speed until light and fluffy. Add persimmon pulp, milk and eggs; beat on medium speed until combined. Add flour mixture and stir until well combined.
Divide batter evenly among paper baking cups (about 3/4 full). Bake 30 minutes until toothpick inserted in center comes out clean. Transfer cupcakes to cooling rack to cool completely.
Meanwhile, make frosting: in large bowl or bowl of clean stand mixer, beat butter until smooth and creamy. Beat in powdered sugar. Stir in milk, ginger and vanilla.
Frost fully cooled cupcakes. Sprinkle with cinnamon, if desired.